Piezoelectric transducer utilizing a catenoidally tapered horn

ABSTRACT

A piezoelectric transducer for converting high power electrical energy to high power mechanical energy at a sonic frequency range. Specifically, the invention is an electromechanical transducer capable of delivering high power levels ( 30 kw) in a high-Q, high efficiency construction utilizing ceramic polycrystalline driving elements. Reference is made to the claims for a legal definition of the invention.

United States Patent [72] Inventor Hildegard M. Minchenko [56]References Cited Reynoldsburg, Ohio UNITED STATES PATENTS g f 333;? 19693,368,085 8/1968 Minchenko 310/82 3,396,8 196 M t :1 1 8. x 45 PatentedJune 29,1971 2 5 8 cMas ere a 3 0/ 2 [73} Assignee The Ohio StateUniversity Duggafl C b Ohi Assistant Examiner-Mark O. BuddAnorneyAnthony D. Cennamo [54] PIEZOELEC'IRIC TRANSDUCER UTILIZING A Tf2 t Z t HORN ABSTRACT: A piezoelectric transducer for converting high an g power electrical energy to high power mechanical energy at a [52]U.S.Cl. BIO/8.2, sonic frequency range. Specifically, the invention isan elec- 310/8.3, 310/87 tromechanical transducer capable of deliveringhigh power [51] Int. Cl H04r 17/00 levels 30 kw) in a high-Q. highefficiency construction [50] Field of Search 310/82, utilizing ceramicpolycrystalline driving elements. Reference 8.3, 8.7, 8.1, 8.0, 9.1 ismade to the claims for a legal definition ofthe invention.

ox (I y I F'\ l i D L x 8 S -u L --w L PATENTEDJUNZQIQTI 3.590.288

sum 1 [IF 2 FIG. I Y

FIG. 2

FIG. 3

1: mm 1': on.

HILDEGARD MINCHENKO ATTORNEY PATENTEUJUNQSIQYI 3.590288 SHEET 2 OF 2||.6 TIM 1 M2 2 no L-.-J

I ma 6 nos 2 5 I04 DIAMETER RATIO FIG. 4

u 5 0 3 a :5 46 g 4.4

LL] 5 36 g DIAMETER RATIO FIG.5 l5

[I 50 ES 665 :2

I INVENTOR.

\ 2 3 4 5 e 7 a 9 HILDEGARDMINCHENKO DIAMETER RATIO FIG. 6 B g ATTORNEYPIEZOELECTRIC TRANSDUCER UTILIZING A CATENOIDALLY TAPERED HORNBACKGROUND An electromechanical transducer such as a piezoelectricdevice is capable of transforming high frequency electrical impulsesinto high frequency mechanical impulses or vice versa. A sonictransducer of this type is disclosed by the present inventor, US. Pat.No. 3,396,285, dated Aug. 6, 1968, for Electromechanical Transducer,assigned to The Ohio State University.

The use of sonic energy has been suggested extensively in all fields ofendeavor. Although the use of sonic energy has been at an increasingpace, realistically its use hasbeen limited by one primary factor, i.e.,power. The prior art, in referring to high-power transducers, refers totransducers up to l watts." For such purposes as earth removal, concretedestruction (road wrecking) and other similar work efforts, this amountof power is negligible. A transducer to be economically acceptable inheavy work loads of this nature should have a power output of ahorsepower or greater.

Until the above disclosure by the inventor, known transducers generallylacked the ability to handle large power levels continuously. Further,the permissible stresses and strains that could be endured by theelectromechanical transducer material were so low that these materialswere soon destroyed by the amounts of power than can be usefullyemployed in such processes. The only realistic solution hereintofore,was the brute force method of employing more transducers oralternatively by increasing the number of horns. The cost of aninstallation of appreciable size is beyond economically acceptablelimits. Simply then, the art awaited a transducer that at a reasonablecost would have a high energy output.

, both radially and longitudinally (axially). In this way the acousticstresses in the piezoelectric elements are always compressive, nevertensile, even under maximum voltage excitation.

SUMMARY The present invention is an improvement of the above-mentionedsonic transducer. The highest power rating that could be achieved in thepast was kw. of sonic energy. The present invention providesapproximately 30 kwf of sonic Another object of the invention is toprovide an electromechanical sonic transducer with an efficiency ofapproximately 97.5 percent.

Another object of the invention is to provide an elec- BRIEF DESCRIPTIONOF THE DRAWINGS FIG. I is a diagrammatic illustration of the crosssection of an acoustic resonator showing longitudinal displacement andstrain relations;

FIG. 2 is a diagrammatic illustration of a cylindrical sandwich typeresonator;

FIG. 3 is a diagrammatic representation of the preferred embodiment ofthe invention;

FIG. 4 is a graphical representation of the resonant length of thepreferred embodiment of FIG. 3 as a function of the diameter ratio for a10.054-kI-Iz. catenoidal horn with a small energy which can be usedeconomically for continuous duty operation in a number of industrialprocesses.

nant length, nodal position, and amplification factor as a function ofdiameter ratio.

OBJECTS Accordingly a principal object of the invention is to provide animproved electromechanical sonic transducer.

Another object of the invention is to provide an electromechanical sonictransducer with minimal losses and a very high-Q.

diameter of 2 inches;

FIG. 5 is a graphical representation of the nodal position with respectto the large diameter of the preferred embodiment of FIG. 3 as afunction of the diameter ratio for a 10.054-

I kHz. catenoidal horn with a small diameter of 2 inches; and,

FIG. 6 is a graphical representation of the amplification factor of thepreferred embodiment of FIG. 3 as a function of the diameter ratio for a10.054-kI-Iz. catenoidal horn with a small diameter of 2 inches.

DESCRIPTION OF THE PREFERRED EMBODIMENT The following analysis providesa complete understanding of the theory underlying the design andoperation of the transducer disclosed herein.

Referring to FIG. 1 the following assumptions are made: (I) the strainis uniformly distributed through the entire cross section, and (2) thewave front remains flat. It can be shown that this approximation isvalid for longitudinal resonators whose length considerably exceeds thediameter.

Throughout the entire derivation, the units of measurement for all termsare given in the MKS (meter-kilogram-second) system. The symbols usedare defined as follows:

x= longitudinal distance along X-axis (m) y radial distancefrom X-axisto the surface of the resonator (m) s= displacement from equilibriumposition (m) s strain (m/m) A cross section area (m F and F =compressiveforce (newtons) Y= Youngs modulus (newton/m) p density of resonatormaterial (kg/m t= time (seconds) o'= longitudinal stress (newtonslm Llength of resonator (m) w=21rf= angular frequency (radians/sec) ffrequency of exciting force (Hz) A wavelength of sound in resonatormaterial (m Referring to FIG. 1, and utilizing the above approximations,only one component s N of the strain tensor will be in existence. Iflongitudinal compressive forces are applied, the plane at-x will bemoved the distance s, the plane at dx the distance s ds. Since dx isvery small the displacement at x dx can be written as:

s+ds =s+(6s/8x)dx. Eq. (1) The increase in length of the segment oflength sx then is Now the strain is defined as the ratio of increase inlength to original length or was; i

dx 6:r Eq. (3)

The longitudinal stress is defined as the compressive force per unitarea:

a=(Force/Area)=(F/A) Eq. 4) Hookes law states that the elastic constantof the material or Young's modulus is equal to the negative ratio ofstress over strain;

Rewriting Eq. (5) yields an expression for the internal longitudinalelastic force:

where A indicates that the cross-sectional area is a function of x.

it should be noted that, in the dynamic case, the elastic forces at thevarious cross sections will differ. If F is the force at x, theF=(8F/6x) represents the force at x=dx. Therefore the incrementalelastic force in the segment of length dx can be expressed as:

Eq. (7) Substituting Eq. (6) into Eq. (7) yields:

a Os a os dF- [-AJ 8i)]dx=Y-az[11, )]d.7c

Siiice the mass of the segment is pA dx, the inertial force ofthesegment can be expressed as where 5/81 is the acceleration of thesegment cit.

From equilibrium considerations, the elastic force increment dF has toequal the inertial force increment dF',

By introducing,( Y/5)=Ac,where c is the velocity of the resonatormaterial, Eq. (lmcan also be written in the form:

as a d. a oz ox A, o, a, Eq. 11 Eq. (ll) is the basic wave or resonatorequation for solid elastic materials.

The basic wave equation is alsoyalid for transmission lines. It isimmaterial whether they are straight-through transmission lines, wherethe incoming signal amplitude is transmitted with a M ratio, ortransmission lines with varying cross sectional area, where the signalamplitude is increased or decreased.

A sonic or ultrasonic force concentrator is a transmission line whichaccepts input energy at one end and delivers it concentrated into asmaller area at its output end.

Under the assumption of a half-wave force generator and a half-waveforce concentrator, and ideal matching conditions, the followingboundary values can be imposed:

at .t=0, strain s'=0 at x=L; strain s'=0.

Under the conditions the concentrator resonates and will not alter themode of operation of the force generator. That means the displacement onthe large end F80, where s is the amplitude of vibration of the forcegenerator.

For the design of a useful force concentrator, various considerationsmust be taken: l) the desired frequency determines the overall length,and (2) the stress and strain distributions along the force concentratorhave to be determined in order to design a horn for a particularapplication and to insure a reasonable life expectancy. Horns with largeamplifica tion factors may physically break up when the stress in thematerial exceeds its fatigue strength. (3) The node locations planes ofzero vibration amplitude in the axial directionmust be determined toprovide locations where support structures can be attached withoutaffecting the acoustical characteristics of the horn. (4) Theamplification factorthe ratio of vibration amplitude at the output endto that at the input endis another factor of importance.

The force concentrator of the present invention is a catenoidal horn.This particular type concentrator has been found to be very effectiveand superior to stepped, conical, or exponential horns for high powerapplications.

in the following, the generating line" is defined as that partieularline, which rotated about the X-axis, will generate the surface underconsideration. Subscripts 0" relate variables to the location x=0 alongthe X-axis and subscripts "L" refer to the location x=L. Also use ismade of the following relation- Referring again to FIG. 1, thegenerating line for a catenoidal horn is given by:

y cosh[m(L-x)] Eq. (13) where the shape parameter m=( l/L) coshR, andthe diameter ratio R=D,,/D,

The area relationship and its derivative for any point along the X-axisare given by:

A,= cosh [m(Lx)] Eq. (14 (A,) 2mA cosh [m(L-:z:)] sinh [m(La:)]

where C and C are constants to be evaluated by imposing the boundaryconditions as stated previously. As a result, the displacement along theresonant catenoidal horn is determined as cosh (mL) cos [b(L-x)] cos(bL) cosh ML-m where b= /'];2: 2 The strain distribution is:

, cosh (mL) sin [b(L-:z:)]

cos (bL) cosh [m(L-:z:)] +m cos [b(L-:c)] tanh [m(L:c)]

cosh [m(La:)]

The frequency equation is:

(bL) tan (bL)= ff5 cosh R Eq. (20) The resonant length is:

where (bL) are the roots of the frequency equation (20). The nodes arelocated at:

(2n-- 1)1r x =L- -,n0, 1, 2 x L (22) The amplification factor M is: i

s L cosh (rnL) R 7 so cos (bL) cos (bL) Eq. (23

The above derivations are based on the thin-rod velocity of thematerial, without taking into account the radial displacements. Actuallythe presence of these displacements will somewhat alter the resonatinglength of the concentrator. One versed in the art can easily derive thenecessary correction factor for small values of R /L and estimate theaccuracy of the above formulas for different values of R /L.

The capability to convert electrical energy into vibratory output energyand the associated efficiency are important parameters in the design ofthe force generator. Therefore. piezoelectric materials are preferredover magnetostrictive materials. However. the fragility of the ceramicelements has to be considered. Only if that particular problem can besolved successfully. can a superior force generator be achieved.

The most efficient use can be made of the ceramic piezoelectric elementsin a Langevin (sandwich) type force generator. The power conversioncapability of the elements can be increased by the application ofhydrostatic compression. However, it is important also to match theimpedance of the ceramic elements to the impedance of the adjacentmaterials. The overall length of the compositeresonator depends upon thedimensions and acoustical properties of the piezoelectric element. Ifthe natural resonance frequency of the unloaded element is known, thelength of the loading sections can be calculated to reduce the frequencyof the system to the desired natural frequency.

For the cylindrical force generator, the result can be summarized asfollows: Referring to FIG. 2. the impedance Z of which should also equalthe magnitude of the impedance 2 of the ceramic element at theinterface, which is wnere: f,, natural resonant frequency of unloadedpiezoelectric element f natural resonant frequency of system.

Subscripts l and c" refer to section I and piezoelectric elementsrespectively.

Equating Z to Z,., the length of sections 1 or 2 can be determined for acertain frequency, or the frequency can be determined for a certainlength.

The driving element of the force generator is of extreme importance. Inthe particular case, a very fragile ceramic constitutes thepiezoelectric element. Due to this fact, conventional transducers havevery low power ratings, unless a multiplicity of individual transducersare paralleled.

The solution to dealing with the fragility of the ceramic material isdisclosed in my prior U.S. Pat. No. 3,396,285.

In theory and practice, the piezoelectric driving elements are underradial and axial pressure to assure that they do not operate in tensioneven under intense sonic action. Significantly, the structural design ofthe transducer that permits the extraordinary power output from thedriving elements resides in a novel method of clamping the piezoelectricelements both radially and longitudinally. In this way, the acousticstresses in the piezoelectric elements are always compressive, nevertensile, even under maximum voltage excitation. At a frequency of kHz.,the power conversion in the ceramic element is approximately twice ashigh as the manufacturers specifications indicate, and yet the elementsdo not destroy themselves even when operated over a long period of time.

Referring now to FIG. 3, there is illustrated the preferred embodimentof the invention. The force concentrator comprises a catenoidal horn 2which is constructed in accordance with the preceding analysis. Thelarge diameter of the horn is 6 inches and the small diameter is 2inches. The 6-inch and 2- inch diameters give the force concentrator adiameter ratio of 3.

The constructed model of the preferred embodiment illustrated in FIG. 3.utilized four rings 8 of piezoelectric material in the force generatorportion 6. These rings 8 and the associated clamping means 4 and conicalsection 50 comprise a half-wave driving structure or force generatorsection of the transducer.

Utilizing the small diameter (2 inches) of the catenoidal horn and theresonant frequency (10.054 kHz.) of the transducer as constantparameters a numerical computer analysis was performed. The resonantlength, nodal position, and amplification factor were determined as abasis of the diameter ratio.

FIG. 4 shows in a graphical manner the resonant length of the transducerof the invention as a function of the diameter ratio. For theconstructed embodiment ratio of 3, it is calculated and is shown thatthe resonant length is l0.0336 inches.

Referring now to FIG. 5, it is seen that the distance from the large endof the concentrator to the nodal plane is plotted as a function of thediameter ratio. A diameter ratio of 3 corresponds with a nodal locationof 3.8939 inches from the force concentrators large diameter.

The graphical plot of the numerical analysis of the amplificationfactor's relation to the diameter ratio is illustrated in FIG. 6. Anamplification factor of 3.57 results when the diameter ratio is chosenas 3.

As can be seen from the graphical representations in FIGS. 4 through 6,for large diameter ratios the relationships are almost linear, while fordiameter ratios between one and three, radical changes in slope arefound to occur.

Although certain and specific embodiments have been illustrated, it isto be understood that modifications may be made without departing fromthe true spirit and scope of the invention.

What I claim is:

l. A high power electromechanical transducer, the improvementcomprising:

a resonant structure including a force generator section, a

force concentrator section, and a plurality of piezoelectric elementspositioned intermediate said generator and said concentrator section ina region near the node of said resonant structure,

said force generator section including an elongated solid portionintegral with said piezoelectric elements and said force concentratorsection,

said force concentrator section comprising a catenoidally tapered barhaving the larger diameter adjacent to said force generator section,

said resonant length and nodal location of said transducer being relatedto its said diameter ratio wherein a diameter ratio of 3 corresponds tol0.03+ inches in length and the nodal location is 3.8+ from the forceconcentrators large diameter.

2. A high power electromechanical transducer as set forth in claim 1wherein the diameter ratios in excess of 3 are linearly related to thelength of said transducer.

3. A high power electromechanical transducer as set forth in claim 1wherein the diameter ratios in excess of 3 are linearly related to thenodal position of said transducer.

4. A high power electromechanical transducer as set forth in claim 1wherein the amplification factor of said transducer is related to saiddiameter ratio wherein the diameter ratio of 3 corresponds to anamplification factor of 3.57.

6. A high power electromechanical transducer as set forth in claim 4wherein the diameter ratios in excess of 3 are linearly related to theamplification factor of said transducer.

UNITED STATES PATENT OFFICE CERTIFICATE OF CORRECTION Patent No. 3,590,288 Dated June 29, 1971 Hildegard M. Minchenko Inventor(s) It iscertified that error appears in the above-identified patent and thatsaid Letters Patent are hereby corrected as shown below:

Column 6, line 51, after "3.8+" insert inches Signed and sealed this25th day of July 1972.

(SEAL) Attest:

EDWARD M.FLETCHER,JR. ROBERT GOTTSCHALK Attesting Officer Commissionerof Patents US. GOVERNMENT FRN'HNG OFFICE: 19.9 6-3i-JSI.

2. A high power electromechanical transducer as set forth in claim 1wherein the diameter ratios in excess of 3 are linearly related to thelength of said transducer.
 3. A high power electromechanical transduceras set forth in claim 1 wherein the diameter ratios in excess of 3 arelinearly related to the nodal position of said transducer.
 4. A highpower electromechanical transducer as set forth in claim 1 wherein theamplification factor of said transducer is related to said diameterratio wherein the diameter ratio of 3 corresponds to an amplificationfactor of 3.57.
 6. A high power electromechanical transducer as setforth in claim 4 wherein the diameter ratios in excess of 3 are linearlyrelated to the amplification factor of said transducer.